Year: 2026
Type: preprint
Abstract: We introduce the analogue, for invariant jet differentials, of the principal symbol of a differential operator: the component of maximal degree in the highest derivative. The top symbol identifies a canonical quotient bundle of the Demailly--Semple bundle $E_{k,m}T^*_X$, with values in jet differentials of lower order, and together with the Brückmann--Rackwitz vanishing theorem it gives a non-ampleness criterion for $E_{k,m}T^*_Z$ on complete intersections $Z\subset\PP^N$. On the Demailly--Semple tower the top symbol is the restriction to a thickening of the vertical divisor, inside the singular jets. The criterion applies in orders $3$ and $4$: $E_{3,5q}T^*_X$ is not ample on any smooth hypersurface $X\subset\PP^4$, and $E_{4,m}T^*_X$ is not ample for $m\ge7$ with $m\equiv0,1,2\pmod 7$ on any smooth hypersurface $X\subset\PP^5$ of degree at least $2$. This contradicts the Diverio--Trapani conjecture at its own threshold.
Source: arXiv (Cornell University)
Author: Diverio, Simone
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214721683